s involving definite integrals (algebraic) AP.CALC: CHA‑4 (EU), CHA‑4.D (LO), CHA‑4.D.1 (EK), CHA‑4.D.2 (EK), CHA‑4.E (LO), CHA‑4.E.1 (EK) Google Classroom Facebook Twitter Email You might need: Calculator Problem The population of a town grows at a rate of $2e^{0.2t}-t$ people per year (where $t$ is the number of years). Approximately by how many people did the population grow between $t=0$ and $t=5$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $5$ (Choice B) B $10$ (Choice C) C $15$ (Choice D) D $20$
Solution: Letting $P(t)$ be the population at year $t$, we are given that $P'(t)=2e^{0.2t}-t$. We want to find $P(5)-P(0)$. According to the Fundamental Theorem of Calculus, $\begin{aligned} P(5)-P(0)&=\int_0^5P'(t)\,dt \\\\ &=\int_0^5(2e^{0.2t}-t)\,dt \end{aligned}$ $\int_0^5(2e^{0.2t}-t)\,dt\approx5$ In conclusion, between $t=0$ and $t=5$ the population grew by $5$ people.